Therefore, the five real non-zero elements of \(\Delta \)  in \(M_p\)  are traded for \(\alpha _L\) , \(\alpha _R\)  and the physical up-type quark masses \(m_u\) , \(m_c\)  and \(m_t\) . In the down sector, on the other hand, both \(U_{L}\)  and \(U_{R}\)  are generic \(3\times 3\)  matrices. While the former is fixed as \(U_{L} = V_L V\) , with the CKM matrix expressed in the Wolfenstein form ( see PDG for details [1]}), the latter can be parameterized with three angles \(\beta _R^{1,2,3}\)  if we assume, as we do in our numerical implementation, that the only complex CP-phase comes from the CKM matrix. This means that, for our scenario, the seven non-zero elements of \(\Gamma \)  in \(M_n\)  can be consistently described with eight real couplings, which are replaced by the four Wolfenstein parameters \(\lambda \) , \(A\) , \(\bar{\rho }\)  and \(\bar{\eta }\) , three quark masses, \(m_d\) , \(m_s\)  and \(m_b\)  and a randomly generated angle which we call \(\beta _R\) . In particular, denoting the solutions of eq:inv for the angles as \(\beta _R^2 = \beta _R^2 (\lambda ,A,\bar{\rho }, \bar{\eta }, m_b, m_s, m_d, \beta _R)\)  and \(\beta _R^3 = \beta _R^3 (\lambda ,A,\bar{\rho }, \bar{\eta }, m_b, m_s, m_d, \beta _R)\) , which are numerically computed, one can express \(U_R\)  as
\(U_R \,=\, \begin{pmatrix}\cos \beta _R \cos \beta _R^2 & \sin \beta _R \cos \beta _R^2 & \sin \beta _R^2 \\-\cos \beta _R \sin \beta _R^2 \sin \beta _R^3 -\sin \beta _R \cos \beta _R^3 & \cos \beta _R \cos \beta _R^3 - \sin \beta _R \sin \beta _R^2 \sin \beta _R^3 & \cos \beta _R^2 \sin \beta _R^3 \\- \cos \beta _R \sin \beta _R^2 \cos \beta _R^3 + \sin \beta _R \sin \beta _R^3 & - \cos \beta _R \sin \beta _R^3 -\sin \beta _R \sin \beta _R^2 \cos \beta _R^3 & \cos \beta _R^2 \cos \beta _R^3 \end{pmatrix} \,,\) 
